Remark 2.3 
The reason to introduce \(\mathcal {C}_m\)  is that \(\mathcal {C}_m\)  is both closed under \(\tau _\mathcal {C}\) -convergence (see [1]}) and \(\mathcal {F}\)  is lower semicontinuous with respect to \(\tau _\mathcal {C}\)  in \(\mathcal {C}_m\)  (see [1]}). Such two properties do not apply instead to \(\mathcal {C}\)  as the following examples show.
